Solved Exercises on Multiples and Patterns in Grade 4

Master multiples and patterns: identifying multiples, recognizing number patterns, continuing sequences, and applying pattern rules through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Identifying multiples
Exercise 1
List the first 8 multiples of 6. What is the pattern in the ones digits?
Definition:

Multiple: The product of a number and any whole number

Pattern: A sequence that repeats in a predictable way

Finding multiples method:
  1. Multiply the base number by 1, 2, 3, 4, etc.
  2. Record each result
  3. Look for patterns in the results
  4. Verify the pattern continues
Base number
6
First 8 multiples
6, 12, 18, 24, 30, 36, 42, 48
Step 1: Multiply 6 by 1 to 8

6 × 1 = 6

6 × 2 = 12

6 × 3 = 18

6 × 4 = 24

6 × 5 = 30

6 × 6 = 36

6 × 7 = 42

6 × 8 = 48

Step 2: Identify the ones digits

6, 2, 8, 4, 0, 6, 2, 8

Step 3: Observe the pattern

The ones digits repeat in the sequence: 6, 2, 8, 4, 0

6
12
18
24
30
36
42
48
Pattern in ones digits:

6, 2, 8, 4, 0, 6, 2, 8

The pattern repeats every 5 multiples: 6, 2, 8, 4, 0

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48
Ones digit pattern: 6, 2, 8, 4, 0
Final answer:

The first 8 multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48

The pattern in the ones digits is: 6, 2, 8, 4, 0 (repeating)

Applied rules:

• Multiple definition: A multiple is the result of multiplying by whole numbers

• Pattern recognition: Look for repeating sequences in digits

• Systematic approach: Multiply in order to find all multiples

2 Continuing number patterns
Exercise 2
Continue the pattern: 5, 10, 15, 20, ___, ___, ___. What is the rule for this pattern?
Definition:

Arithmetic sequence: A sequence where each term increases by a constant amount

Pattern rule: The mathematical operation that generates the sequence

Given sequence
5, 10, 15, 20
Pattern rule
Add 5 each time
Continued sequence
25, 30, 35
Step 1: Identify the difference between terms

10 - 5 = 5

15 - 10 = 5

20 - 15 = 5

Step 2: Determine the pattern rule

Add 5 to each term to get the next term

Step 3: Continue the pattern

20 + 5 = 25

25 + 5 = 30

30 + 5 = 35

5
10
15
20
25
30
35
5
10
15
20
+
25
30
35
Pattern Rule:

Start with 5, add 5 to each term

Or: Each term is 5 times its position (5n)

Continued pattern: 5, 10, 15, 20, 25, 30, 35
Rule: Add 5 each time
Final answer:

The continued pattern is: 5, 10, 15, 20, 25, 30, 35

The rule is: Add 5 to each term to get the next term.

Applied rules:

• Arithmetic sequence: Each term differs by a constant amount

• Difference method: Find the constant difference between terms

• Pattern continuation: Apply the rule to extend the sequence

3 Geometric patterns
Exercise 3
Find the next three terms in this pattern: 2, 6, 18, 54, ___, ___, ___. What is the rule?
Definition:

Geometric sequence: A sequence where each term is multiplied by the same number

Common ratio: The number each term is multiplied by

Given sequence
2, 6, 18, 54
Common ratio
Multiply by 3
Next three terms
162, 486, 1458
Step 1: Find the relationship between terms

6 ÷ 2 = 3

18 ÷ 6 = 3

54 ÷ 18 = 3

Step 2: Identify the pattern rule

Multiply each term by 3 to get the next term

Step 3: Continue the pattern

54 × 3 = 162

162 × 3 = 486

486 × 3 = 1458

2
6
18
54
162
486
1458
2
×
3
6
×
3
18
×
3
54
Pattern Rule:

Start with 2, multiply each term by 3

Common ratio: 3

Continued pattern: 2, 6, 18, 54, 162, 486, 1458
Rule: Multiply by 3 each time
Final answer:

The next three terms are: 162, 486, 1458

The rule is: Multiply each term by 3 to get the next term.

Applied rules:

• Geometric sequence: Each term is multiplied by the same number

• Common ratio: Found by dividing consecutive terms

• Multiplicative pattern: Terms grow by multiplication

Multiples and Patterns Fundamentals
\(\text{Multiple of } n = n \times k \text{ where } k \text{ is a whole number}\)
Multiple Definition
Key definitions:

Multiple: The result of multiplying a number by a whole number (0, 1, 2, 3...)

Pattern: A sequence that follows a specific rule or relationship

Arithmetic sequence: Each term increases by a constant amount

Geometric sequence: Each term is multiplied by a constant amount

Term: Each number in a sequence

Rule: The mathematical operation that generates the sequence

Common difference: The constant amount added in arithmetic sequences

Common ratio: The constant amount multiplied in geometric sequences

Pattern identification methods:
  1. Arithmetic patterns: Find the common difference by subtracting consecutive terms
  2. Geometric patterns: Find the common ratio by dividing consecutive terms
  3. Look for patterns in digits: Observe ones, tens, or hundreds places
  4. Check for multiples: See if terms are multiples of a specific number
  5. Verify the rule: Apply the rule to generate the sequence
Tip 1: For arithmetic sequences, subtract consecutive terms to find the common difference.
Tip 2: For geometric sequences, divide consecutive terms to find the common ratio.
Tip 3: Look for patterns in the digits of multiples (like in the 6s example).
Tip 4: Always verify your pattern rule by applying it to generate the sequence.
Common errors: Mixing up arithmetic and geometric patterns, miscalculating differences/ratios.
Key insight: Multiples form arithmetic sequences with the base number as the common difference.
Pattern rules:

• Arithmetic sequence: nth term = first term + (n-1) × common difference

• Geometric sequence: nth term = first term × (common ratio)^(n-1)

• Multiples: Multiples of n are n, 2n, 3n, 4n, ...

• Even numbers: Multiples of 2 (2, 4, 6, 8, ...)

• Odd numbers: Numbers that are not multiples of 2

Pattern Recognition Strategies

Arithmetic Patterns

Example: 3, 7, 11, 15, 19...

Common difference: 4

Rule: Add 4 each time

Geometric Patterns

Example: 2, 6, 18, 54, 162...

Common ratio: 3

Rule: Multiply by 3 each time

Solution: Exercises 4 to 5
4 Mixed pattern recognition
Exercise 4
The sequence 3, 6, 12, 24, 48... continues. What are the next two terms? Is this an arithmetic or geometric sequence?
Definition:

Sequence type: Whether a sequence is arithmetic or geometric

Given sequence
3, 6, 12, 24, 48
Type of sequence
Geometric
Next two terms
96, 192
Step 1: Check if it's arithmetic (constant difference)

6 - 3 = 3

12 - 6 = 6

24 - 12 = 12

Differences are not constant, so it's not arithmetic

Step 2: Check if it's geometric (constant ratio)

6 ÷ 3 = 2

12 ÷ 6 = 2

24 ÷ 12 = 2

48 ÷ 24 = 2

Ratio is constant (2), so it's geometric

Step 3: Continue the geometric pattern

48 × 2 = 96

96 × 2 = 192

3
6
12
24
48
96
192
3
×
2
6
×
2
12
×
2
24
×
2
48
Pattern Type:

This is a geometric sequence with common ratio 2

Rule: Multiply each term by 2

Next two terms: 96, 192
Type: Geometric sequence
Final answer:

The next two terms are 96 and 192.

This is a geometric sequence with a common ratio of 2.

Applied rules:

• Sequence identification: Check both difference and ratio

• Geometric sequence: Each term is multiplied by the same number

• Verification: Confirm the pattern holds for all terms

5 Advanced pattern application
Exercise 5
Which of these numbers are multiples of 8: 24, 32, 45, 56, 63, 72? How do you know?
Definition:

Multiple check: A number is a multiple of n if it can be divided by n with no remainder

Numbers to check
24, 32, 45, 56, 63, 72
Multiples of 8
24, 32, 56, 72
Step 1: Check each number for divisibility by 8

24 ÷ 8 = 3 (remainder 0) → Multiple of 8

32 ÷ 8 = 4 (remainder 0) → Multiple of 8

45 ÷ 8 = 5.625 (remainder) → Not a multiple of 8

56 ÷ 8 = 7 (remainder 0) → Multiple of 8

63 ÷ 8 = 7.875 (remainder) → Not a multiple of 8

72 ÷ 8 = 9 (remainder 0) → Multiple of 8

Step 2: Verify using multiplication

8 × 3 = 24 ✓

8 × 4 = 32 ✓

8 × 7 = 56 ✓

8 × 9 = 72 ✓

Step 3: List the multiples

24, 32, 56, 72 are multiples of 8

24 ✓
32 ✓
45 ✗
56 ✓
63 ✗
72 ✓
Multiples of 8:

24, 32, 56, 72

These can all be divided by 8 with no remainder

Multiples of 8: 24, 32, 56, 72
Final answer:

The multiples of 8 are: 24, 32, 56, and 72.

I know because each of these numbers can be divided by 8 with no remainder.

Applied rules:

• Divisibility test: A number is a multiple of n if n divides it evenly

• Multiplication verification: Check by multiplying the base number

• Systematic checking: Test each number individually

Advanced Patterns and Multiples
\(\text{nth term of arithmetic sequence} = a_1 + (n-1)d\)
Arithmetic Sequence Formula
Key definitions:

Arithmetic sequence: A sequence where each term increases by a constant difference

Geometric sequence: A sequence where each term is multiplied by a constant ratio

Recursive pattern: A pattern where each term depends on previous terms

Explicit formula: A formula that calculates any term directly

Fibonacci sequence: Each term is the sum of the two preceding terms

Triangular numbers: Numbers that form triangles (1, 3, 6, 10, 15...)

Square numbers: Numbers that form squares (1, 4, 9, 16, 25...)

Cubic numbers: Numbers that form cubes (1, 8, 27, 64...)

Advanced pattern recognition methods:
  1. Look for arithmetic progression: Check for constant differences
  2. Look for geometric progression: Check for constant ratios
  3. Check for special sequences: Fibonacci, triangular, square numbers
  4. Examine digit patterns: Look for repeating digits or patterns
  5. Consider multiple operations: Patterns may involve more than one operation
Tip 1: For complex patterns, look at the differences between terms of the differences.
Tip 2: Square numbers follow the pattern: 1, 4, 9, 16, 25... (n²).
Tip 3: Triangular numbers follow the pattern: 1, 3, 6, 10, 15... (n(n+1)/2).
Tip 4: Always check if the pattern alternates or involves multiple operations.
Common patterns: Multiples, even/odd numbers, powers, factorials.
Pattern complexity: Patterns can combine arithmetic and geometric elements.
Advanced relationships:

• Arithmetic sequence: aₙ = a₁ + (n-1)d where d is the common difference

• Geometric sequence: aₙ = a₁ × r^(n-1) where r is the common ratio

• Triangular numbers: Tₙ = n(n+1)/2

• Square numbers: Sₙ = n²

• Cubic numbers: Cₙ = n³

• Fibonacci sequence: Fₙ = Fₙ₋₁ + Fₙ₋₂

Special Number Sequences

Square Numbers

1, 4, 9, 16, 25, 36, 49, 64, 81, 100...

Formula: n²

Pattern: Differences are consecutive odd numbers

Triangular Numbers

1, 3, 6, 10, 15, 21, 28, 36, 45, 55...

Formula: n(n+1)/2

Pattern: Form triangles with dots

Questions & Answers

Question: I'm having trouble telling the difference between arithmetic and geometric patterns. How can I remember which is which?

Answer: Here's a simple way to remember:

  • Arithmetic patterns: You ADD the same number each time (think "Add")
  • Geometric patterns: You MULTIPLY by the same number each time (think "Multiply")

To identify which type you have:

  • For arithmetic: Subtract consecutive terms - if the difference is always the same, it's arithmetic
  • For geometric: Divide consecutive terms - if the quotient is always the same, it's geometric

Examples:

  • Arithmetic: 5, 8, 11, 14... (add 3 each time)
  • Geometric: 5, 15, 45, 135... (multiply by 3 each time)

The key is to perform the opposite operation of what's happening in the pattern to identify the type.

Question: My child is struggling with finding multiples. What strategies can help them identify multiples quickly?

Answer: Here are effective strategies for identifying multiples:

  • Division test: A number is a multiple of n if it divides evenly by n (no remainder)
  • Counting by n: Skip count by the number (e.g., for multiples of 6: 6, 12, 18, 24...)
  • Factor test: n must be a factor of the number
  • Patterns in digits: Learn the patterns for specific multiples (e.g., multiples of 5 end in 0 or 5)

For specific numbers:

  • Multiples of 2: Even numbers
  • Multiples of 5: End in 0 or 5
  • Multiples of 10: End in 0
  • Multiples of 3: Sum of digits is divisible by 3

Practice with visual aids like number grids where multiples of a number are highlighted. This helps students see the patterns visually.

Question: How do patterns and multiples connect to other mathematical concepts students will encounter?

Answer: Patterns and multiples are foundational to many advanced concepts:

  • Algebra: Understanding sequences prepares students for linear and exponential functions
  • Number theory: Multiples lead to concepts of least common multiple (LCM) and greatest common factor (GCF)
  • Fractions: Finding common denominators relies on multiples
  • Algebraic expressions: Recognizing patterns helps factor polynomials
  • Functions: Arithmetic sequences are linear functions; geometric sequences are exponential functions
  • Coordinate geometry: Patterns can be plotted as points on a coordinate plane

Pattern recognition also develops critical thinking and problem-solving skills. Students who are fluent with multiples and patterns can more easily tackle complex problems that involve identifying relationships between numbers.

Furthermore, understanding arithmetic and geometric sequences provides the foundation for calculus concepts like series and convergence.